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Structural Balance and Partitioning Signed Graphs

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Structural Balance and Partitioning Signed Graphs Structural Balance and Partitioning Signed Graphs Patrick Doreian Andrej Mrvar Department of Sociology Faculty of Social Sciences University of Pittsburgh University of Ljubljana Abstract Signed graphs have been used frequently to model affect ties for social actors. In this context, structural balance theory has been used as a theory for the organization and form of these relations. Based on a graph theoretical formalization of this theory, for signed graphs, there are theorems stating that balanced structures have clear parti- tioned forms. We discuss the use of local optimization procedures as a method of obtaining these partitioned structures – or partitioned structures that are as close to a balanced partition as possible for structures that are not balanced. In addition, this method provides a measure of the extent to which a signed graph is imbalanced. For a time series of signed graphs, this permits a test of the balance hypoth- esis that human groups tend towards balance over time. Examples are provided of interpersonal ties in networks and institutionalized rela- tions among collective units. Since the pioneering work of Heider (1946) there has been an enduring concern with theories of structural balance. See, for example, Taylor (1970) and Forsythe (1990). Central to this formulation is the idea of triads made up of two actors, p and q, and some object, x. There is a direct social relation between the two actors and there are unit formation links between the actors and the object. The set of such triads is displayed in Figure 1. The relation between the actors can be positive (for example, liking) or negative (disliking) and the unit formation relations can be positive (for example, approve) or negative (disapprove). 1 X X X X P O P O P O P O Four Balanced Triadic Configurations X X X X P O P O P O P O Four Imbalanced Triadic Configurations Figure 1: Triadic Configurations. Definition 1 A triad is balanced if the product of the signs that it contains is positive and is unbalanced when the product of its signs is negative. The structures in the top row of Figure 1 are all balanced while those in the second row are imbalanced.1 Heider’s key substantive insight is that people prefer balanced configurations to those that are imbalanced. For Heider, x can be another person, an event, an idea or a thing. Of interest here is the formalization of Heider’s version of this theory by Cartwright and Harary (1956). In their formalization, the distinction between direct affect ties and unit formation ties was deleted – a reasonable step if the ’object’ can include other people. Indeed, the most frequent use of structural balance ideas is when p, q and x are all social actors – in which case, o is used as a label for the third social actor. 1As we use them, the terms imbalanced and unbalanced are interchangeable. 2 1 Signed Graphs and Structural Balance Cartwright and Harary (1956) used graphs and digraphs to formalize struc- tural balance ideas. Definition 2 A signed digraph is an ordered pair, (G, σ), where (Batagelj, 1994:56): • G =(V, R) is a digraph, without loops, having a set of vertices, V , and a set of arcs, R ⊑ V × V ; and • σ : R →{+1, −1} is a sign function. The positive arcs are assigned to +1 while the negative arcs are assigned to −1. Clearly, such an assignment can be made for each of the triads shown in Figure 1. However, Cartwright and Harary extended the consideration of triads to include cycles and semicycles of any length. Definition 3 A cycle (semicycle) is positive if the product of the signs of its edges (arcs) is positive and negative if the product of the signs of its edges (arcs) is negative. Definition 4 A signed graph (digraph) is balanced if all of its cycles (semi- cyles) are positive. In terms of these definitions, Cartwright and Harary (1956: 286) proved the, so-called, structure theorem: Theorem 1 If a signed graph (digraph) is balanced, then the set of vertices, V , can be partitioned into two subsets, called plus-sets, such that all of the positive edges (arcs) are between vertices within a plus-set and all negative edges (arcs) are between vertices in different plus-sets. This result is illustrated on the left side of Figure 2 where the graph on the left is balanced while the graph on the right is not. On the left, the plus sets are {a, c} and {b, d}. The positive edge between a and c links vertices in {a, c} and the other positive edge, between b and d, links vertices within {b, d}. The negative edges, between a and b and between c and d, link vertices 3 a b a b a b f g e c d c d 2-Balanced Not 2-Balanced c d h i Figure 2: Two Undirected Signed Graphs (left) and A 3-Balanced Undirected Signed Graph (right). in different plus sets. For the graph on right, the presence of the positive edge between a and d, means that the graph is not balanced. The triads {a, b, d} and {a,c,d} are both negative and there is no partition consistent with the structure theorem. Consider next the graph shown on the right side of Figure 2, one that is not balanced in the above sense. In terms of triads (or 3-cycles) it is clear that {b, e, f} and {d, e, h} are both negative. Thus there is no partition into two plus-sets that is consistent with Theorem 1. Yet, viewing the structure, there is a clear partition. There is one subset, {a,b,c,d}, and another subset, {f,g,h,i}, that are plus-sets with only negative edges between them. And both of these subsets have negative edges with e. It seems reasonable, then, to think in terms of partitioning the vertices of a graph into more than two plus-sets. (In the imagery of human groups, the graph on the right side of Figure 2 describes two subgroups with internal positive ties and external negative ties plus another individual that is disliked by some members of each of the subgroups.) For the classical concept of structural balance there are two plus sets and we can label this as 2-balance. For the graph on the right side of Figure 2 we can talk of 3-balance where there are 3 plus-sets where all of the positive lines are within plus-sets and all of the negative lines are between plus-sets. This can be extended to a concept of k-balance in an obvious way with k the number of plus-sets in the partition of nodes. In effect, Davis (1967: 183) pursued this line of thought and established: Theorem 2 A signed graph (digraph) is k-balanced if and only if it contains no semicycles with exactly one negative edge (arc). 4 For structural balance, the all negative triad is defined as imbalanced. Davis argued that it is just as reasonable to declare this as balanced. When this done, the structure Theorem 1 becomes the more general structure the- orem 2. We label ’balance’ for Theorem 2 as ’generalized balance’ (i.e. for k > 2) and reserve ’structural balance’ for k = 2. The psychological balance theories claim that the (network) structures are either balanced or move towards a state of balance. For the latter case, measures of imbalance are needed if we want to trace movement towards balance. Two types of measures exist. One is couched in terms of cycles while the other is defined in terms of edges (or arcs). For cycles, if C(G) is the total number of cycles in G and C+(G) is the number of positive C+(G) cycles, then imbalance is measured as 1 − C(G) (Cartwright and Harary, 1956: 288). Variations of this basic definition consider also the length of the cycles. Cycles beyond a certain length can be ignored or cycles can have (inverse) weights depending on their lengths. If we focus attention on edges (arcs), another type of measure of imbalance is the number of links that are inconsistent with the balance partition. For structural balance (Harary et al., 1965: 348), it is the number of links, imb, that have to change sign in order to create a balanced graph (and sign changes for all proper subsets of the links contributing to imb do not do so.)2 We will use imb as a measure of imbalance for both structural and generalized balance. 2 Partitioning Signed Graphs The structure theorems are stated for signed graphs that are balanced (for either structural or generalized balance). They are existance theorems and are silent about the composition of the plus-sets. For imbalanced graphs they have no relevance. Yet most empirical structures are not balanced. Doreian and Mrvar (1995) have proposed a method for obtaining clusters as close to balance (in either sense) as possible together with a minimized value of criterion function stated in terms of imb. Let N be the number of negative ties (edges or arcs) that link directly vertices in the same plus-set. Similarly, let P be the number of positive 2Another definition of imbalance is the number of edges that must be removed in order to create a balanced graph and is termed an edge-deletion measure. Harary et al. (1965) prove that the edge-deletion measure is the same as the sign change measure. 5 ties that link vertices in different plus-sets.
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